SHARP ERROR ESTIMATE OF BDF2 SCHEME WITH VARIABLE TIME STEPS FOR LINEAR REACTION-DIFFUSION EQUATIONS
Jiwei Zhang, Chengchao Zhao
Abstract
While the variable time-steps two-step backward differentiation formula (BDF2) is valuable and widely used to capture the multi-scale dynamics of model solutions, the stability and convergence of BDF2 with variable time steps still remain incomplete. In this work, we revisit BDF2 scheme for linear diffusion-reaction problem. By using the technique of the discrete orthogonal convolution (DOC) kernels developed in [ 11 ], and introducing the concept of the discrete complementary convolution (DCC) kernels, we present that BDF2 scheme is unconditionally stable under a adjacent time-step ratio condition: 0 < rk: =τk/τk-1 ≤ rmax ≈ 4.8645. With the uses of DOC and DCC kernels, the second-order temporal convergence can be achieved under 0 < rk ≤ rmax ≈ 4.8645. Our analysis shows that the second-order convergence is sharp and robust. The robustness means that the second-order convergence is sharp for any time step satisfying 0 < rk ≤ rmax ≈ 4.8645, this is, it does not need extra restricted conditions on the time steps. In addition, our analysis also shows that the first level solution u1 obtained by BDF1 (i.e. Euler scheme) does not cause the loss of global accuracy of second order with 0 < rk ≤ 4.8645. Numerical examples are provided to demonstrate our theoretical analysis.